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Rates of consistency for nonparametric estimation of the mode in absence of smoothness assumptions

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  • Herrmann, Eva
  • Ziegler, Klaus
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    Abstract

    Nonparametric estimation of the mode of a density or regression function via kernel methods is considered. It is shown that the rate of consistency of the mode estimator can be determined without the typical smoothness conditions. Only the uniform rate of the so-called stochastic part of the problem together with some mild conditions characterizing the shape or "acuteness" of the mode influence the rate of the mode estimator. In particular, outside the location of the mode, our assumptions do not even imply continuity. Overall, it turns out that the location of the mode can be estimated at a rate that is the better the "peakier" (and hence nonsmooth) the mode is, while the contrary holds with estimation of the size of the mode.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 68 (2004)
    Issue (Month): 4 (July)
    Pages: 359-368

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    Handle: RePEc:eee:stapro:v:68:y:2004:i:4:p:359-368

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    Keywords: Nonparametric curve estimation Mode Kernel smoothing Rates of consistency Nonsmooth curves;

    References

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    1. Vieu, Philippe, 1996. "A note on density mode estimation," Statistics & Probability Letters, Elsevier, vol. 26(4), pages 297-307, March.
    2. Joseph Romano, 1988. "Bootstrapping the mode," Annals of the Institute of Statistical Mathematics, Springer, vol. 40(3), pages 565-586, September.
    3. Liebscher E., 2001. "Estimation Of The Density And The Regression Function Under Mixing Conditions," Statistics & Risk Modeling, De Gruyter, vol. 19(1), pages 9-26, January.
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    Cited by:
    1. Shi, Xiaoping & Wu, Yuehua & Miao, Baiqi, 2009. "A note on the convergence rate of the kernel density estimator of the mode," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1866-1871, September.
    2. Salah Khardani & Mohamed Lemdani & Elias Ould Saïd, 2012. "On the strong uniform consistency of the mode estimator for censored time series," Metrika, Springer, vol. 75(2), pages 229-241, February.
    3. Obereder, Andreas & Scherzer, Otmar & Kovac, Arne, 2007. "Bivariate density estimation using BV regularisation," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5622-5634, August.

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